Let $X$ be a (finite if you want) set and form the simplicial set $F^{\bullet}(X)$ with $$ F^{n}(X) = \mathrm{Hom}_{\mathrm{set}} ([n], X) $$ where the right hand side denotes arbitrary maps of sets (of course it wouldn't make sense to say order preserving as $X$ doesn't come with an order).

I'm wondering about a description of $F^{\bullet}(X)$. For example if $X = \{0,1\}$ then there are 2 0-simplices, may as well call them $[0] and [1]$ and 2 1-simplices $[0, 1]$ and $[1,0]$ glued together to form a copy of $S^1$.

Edit: as pointed out in Goodwillie's answer, this is not the end of the story, there are way more higher dimensional non-degenerate simplices.

Is there an analogous description when $X = \{0, 1, 2\}$?

A closely related question is whether there's a right adjoint to the forgetful functor from the simplex category $\Delta$ (finite ordered sets) to, say, finite (unordered) sets -- and if so what is it.

Example where such simplicial sets arise: given a map of topological spaces $f: X \to Y$ we can always form a simplicial object $\mathcal{S}^{\bullet}(f)$ with $$ \mathcal{S}^{n} = \prod\nolimits_{X}^{n} = \underbrace{X \times_{Y} \cdots \times_{Y} X}_{n\text{ times }} $$ with face and degeneracy maps given by projections and diagonals respectively. Taking connected components gives a simplicial set.

When $Y$ is the union $\bigcup_{i=1}^{N} H_{i}$ of the coordinate hyperplanes in $\mathbb{C}^{N}$ and $f: X=\coprod_{i=1}^{N} H_{i} \to \bigcup_{i=1}^{N} H_{i}=Y$ is the obvious map, I believe the simplicial set we get is $F^{\bullet}(\{1, \dots, n\})$.


You are overlooking some nondegenerate simplices. For example, when $X={0,1}$ there are the $2$-cells $[0,1,0]$ and $[1,0,1]$. In fact, the thing you call $F^\bullet(X)$ is infinite dimensional if $X$ has more than one element.

It is contractible whenever $X$ is non-empty; this can be seen by identifying it with the nerve of a category, a category equivalent to the point category with one morphism.

If $X=G$ has a group structure then $F^\bullet(G)$ is often called $EG$; it is a contractible space with free $G$-action.

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    $\begingroup$ Ah! Absolutely overlooked those, many thanks. Quick follow up: as an intermediate step, can we view $F^\bullet(X)$ as the nerve of the category with objects the points $x \in X$ and with a unique morphism $x \to y$ for every 2 points $x, y \in X$ (this is at least the case when $X= G$ a group and we build $EG$ as the nerve of the Cayley graph I think...)? Then including a point $x \in X$ with its identity morphism would give an equivalence of categories ... $\endgroup$ – cgodfrey Apr 29 at 2:26
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    $\begingroup$ Yes. That's what I meant in my second paragraph! $\endgroup$ – Tom Goodwillie Apr 29 at 20:55

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